Initially, mesh vertices are simplified by treating them as a point cloud but, instead of using a k-nn, the mesh adjacency matrix is utilized in order to determine point connectivity.
In a second step, the remaining vertices are re-triangulated using an off-the-self triangulation algorithm such as Ball Pivoting [1]}. Different triangulation algorithms, such as Delaunay, alpha shapes or Voronoi diagrams, could be also used but we observed that Ball Pivoting algorithm produces better results in small point clouds. Figure REF  shows the extension and application of the proposed simplification method to triangular meshes (for various simplification ratios).
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